ALBERT

Description: The trajectory of a non-isolated monopole on the beta-plane is calculated as an asymptotic expansion in the ratio of the strength of the vortex to the beta-effect. The method of matched asymptotic expansions is used to solve the equations of motion in two regions of the flow: a near field where the beta-effect enters as a first-order forcing in relative vorticity, and a wave field in which the dominant balance is a linear one between the beta-effect and the rate of change of relative vorticity. The resulting trajectory is computed for Gaussian and Rankine vortices.

Description: This paper formulates a model of mixing in a stratified and turbulent fluid. The model uses the horizontally averaged vertical buoyancy gradient and the density of turbulent kinetic energy as variables. Heuristic 'mixing-length' arguments lead to a coupled set of parabolic differential equations. A particular form of mechanical forcing is proposed; for certain parameter values the relationship between the buoyancy flux and the buoyancy gradient is non-monotonic and this leads to an instability of equilibria with linear stratification. The instability results in the formation of steps and interfaces in the buoyancy profile. In contrast to previous ones, the model is mathematically well posed and the interfaces have an equilibrium thickness that is much larger than that expected from molecular diffusion. The turbulent mixing process can take one of three forms depending on the strength of the initial stratification. When the stratification is weak, instability is not present and mixing smoothly homogenizes the buoyancy. At intermediate strengths of stratification, layers and interfaces form rapidly over a substantial interior region bounded by edge layers associated with the fluxless condition of the boundaries. The interior pattern subsequently develops more slowly as interfaces drift together and merge; simultaneously, the edge layers advance inexorably into the interior. Eventually the edge layers meet in the middle and the interior pattern of layers is erased. Any remaining structure subsequently decays smoothly to the homogeneous state. Both the weak and intermediate stratified cases are in agreement with experimental phenomenology. The model predicts a third case, with strong stratification, not yet found experimentally, where the central region is linearly stable and no steps form there. However, the edge layers are unstable; mixing fronts form and then erode into the interior.

Description: The scattering of acoustic waves by compact three-dimensional axisymmetric vortices is studied using direct numerical simulation in the case where the incoming wave is aligned with the symmetry axis and the direction of propagation of the vortices. The cases of scattering by Hill's spherical vortex and Gaussian vortex rings are examined, and results are compared with predictions obtained by matched asymptotic expansions and the Born approximation. Good agreement is obtained for long waves, with the Born approximation usually giving better predictions, especially as the difference in scale between vortex and incoming waves decreases and as the Mach number of the flow increases. An improved version of the Born approximation which takes into account higher-order effects in Mach number gives the best agreement.

Description: We obtain an analytic solution for the generation of internal gravity waves by tidal flow past a vertical barrier of height b in a uniformly stratified ocean of depth h 〉 b and buoyancy frequency N. The radiated power (watts per metre of barrier) is 1/4πρ0b2U2N√1 - (f/ω 2M(b/h), where ρ0 is the mean density of seawater, U cos(ωt) the tidal velocity, and f the Coriolis frequency. The function M(b/h) increases monotonically with M(0) = 1, M(0.92) = 2 and M(1) = ∞. As b/h → 1, M diverges logarithmically and consequently the radiated power grows as In[(h - b /b]. We also calculate the conversion in a realistically stratified ocean with strongly non-uniform buoyancy frequency, N(z). A rough approximation to the radiated power in this case is 1/4πρ0b2U2N(b)√1 - (f/ω)2M(b/π), where N(b) is the buoyancy frequency at the tip of the ridge and B is the height of the ridge in WKB coordinates. (The WKB coordinate is normalized so that the total depth of the ocean is π.) The approximation above is an over-estimate of the actual radiation by as much as 20% when B/π ≈ 0.8. But the formula correctly indicates the strong dependence of conversion on stratification through the factor N(b).

Description: A two-dimensional model for the flapping of an elastic flag under axial flow is described. The vortical wake is accounted for by the shedding of discrete point vortices with unsteady intensity, enforcing the regularity condition at the flag's trailing edge. The stability of the flat state of rest as well as the characteristics of the flapping modes in the periodic regime are compared successfully to existing linear stability and experimental results. An analysis of the flapping regime shows the co-existence of direct kinematic waves, travelling along the flag in the same direction as the imposed flow, and reverse dynamic waves, travelling along the flag upstream from the trailing edge. © 2008 Cambridge University Press.

Description: We investigate the scattering of a plane acoustic wave by an axisymmetric vortex in two dimensions. We consider vortices with localized vorticity, arbitrary circulation and small Mach number. The wavelength of the acoustic waves is assumed to be much longer than the scale of the vortex. This enables us to define two asymptotic regions: an inner, vortical region, and an outer, wave region. The solution is then developed in the two regions using matched asymptotic expansions, with the Mach number as the expansion parameter. The leading-order scattered wave field consists of two components. One component arises from the interaction in the vortical region, and lakes the form of a dipolar wave. The other component arises from the interaction in the wave region. For an incident wave with wavenumber k propagating in the positive X-direction, a steepest descents analysis shows that, in the far-field limit, the leading-order scattered field takes the form i(π - θ)eikX + 1/2 cos θ cot (1/2θ)(2π/kR)1/2 ei(kRπ/4), where θ is the usual polar angle. This expression is not valid in a parabolic region centred on the positive X-axis, where kRθ2 = O(1). A different asymptotic solution is appropriate in this region. The two solutions match onto each other to give a leading-order scattering amplitude that is finite and single-valued everywhere, and that vanishes along the X-axis. The next term in the expansion in Mach number has a non-zero far-field response along the X-axis.

Description: An analytical method for determining the shape of hollow vortices in shear flows is presented in detail. In a non-dimensional formulation, it is shown that the problem has one degree of freedom represented by the free choice of the non-dimensionalized speed at the boundary of the vortex. The solutions form two families of shapes corresponding to vortex circulation and shear-flow vorticity having the opposite or same sign. When the signs are opposite, the shape family resembles that described by Llewellyn Smith & Crowdy (J. Fluid Mech., vol. 691, 2012, pp. 178-200) for hollow vortices in a potential flow with strain. As for that flow, there is a minimum value of below which there is no solution as the boundary of the vortex self-intersects, while, when the signs are the same, solutions exist for 〈![CDATA[$0. © 2016 Cambridge University Press.

Description: We examine the dynamics of a semi-infinite vortex sheet attached not to a semi-infinite plate but instead to a rigid right-angled wedge, with the sheet aligned along one of its edges. Our approach to this problem, which was suggested by David Crighton, accords well with the fundamental ethos of Crighton's work, which was characterized by 'the application of rigorous mathematical approximations to fluid mechanical idealizations of practically relevant problems' (Ffowcs Williams, Annu. Rev. Fluid Mech., vol. 34, 2002, pp. 37-49). The resulting linearised unsteady potential flow is forced by an oscillatory dipole in the uniform stream passing along the top of the wedge, while there is stagnant fluid in the remaining quadrant. Spatial instability is considered according to well-established methods: causality is enforced by allowing the frequency to become temporarily complex. The essentially quadrant-type geometry replaces the usual Wiener-Hopf technique by the Mellin transform. The core difficulty is that a first-order difference equation of period 4 requires a solution of period unity. As a result, the complex fourth roots of appear in the complementary function. The Helmholtz instability wave is excited and requires careful handling to obtain explicit results for the amplitude of the instability wave. © © 2016 Cambridge University Press.

Description: We extend the vorticity-based modelling approach of Borden & Meiburg (Phys. Fluids, vol. 25 (10), 2013, 101301) to non-Boussinesq gravity currents and derive an analytical expression for the Froude number without the need for an energy closure or any assumptions about the pressure. The Froude-number expression we obtain reduces to the correct form in the Boussinesq limit and agrees closely with simulation data. Via detailed comparisons with simulation results, we furthermore assess the validity of three key assumptions underlying both our as well as earlier models: (i) steady-state flow in the moving reference frame; (ii) inviscid flow; and (iii) horizontal flow sufficiently far in front of and behind the current. The current approach does not require an assumption of zero velocity in the current. © 2016 Cambridge University Press.